Identification Methods for Structural Systems

Transcription

1 Prof. Dr. Eleni Chatzi Lecture May, 2013 Courtesy of Prof. S. Fassois & Dr. F. Kopsaftopoulos, SMSA Group, University of Patras

2 Statistical methods for SHM courtesy of Prof. S. Fassois & Dr. F. Kopsaftopoulos, SMSA Group, UPatras Overview Introduction Non-parametric methods Parametric methods Fault Detection and Identification Application Study

3 Introduction SHM Concept Fault detection (presence of fault) Faut identification (type location) Fault estimation (fault size) Remaining life estimation

4 Introduction Properties of Statistical SHM methods Global methods working at a system level Time and cost effective Automation capability No visual inspection Adaptable to on-line use Random vibration is commonly encountered no need for special experimental conditions/procedures Less sensitive than certain local methods Application examples In-flight aircraft monitoring Bridge and buildings under earthquake excitation Surface vehicle monitoring Underwater, sea vessel, off-shore platform monitoring Railway bridge monitoring Monitoring of large industrial structures

5 Introduction Advantages Inherently optimal accounting of uncertainty and random vibration Accounting for exogenous uncertainties (measurement errors, environmental factors, boundary conditions,...) No requirement for physical or finite element models No requirement for complete models (partial models may be used) Optimal statistical decision making under uncertainty Pitfalls They may locate faults only to the extent allowed by the type of model used Baseline phase requires groundwork under various damage types

6 Introduction Precise problem statement Structure in currently unknown state (S u). May be healthy (S o) of faulty ( S o). If faulty, it will belong to a specific fault mode/type A,..., D (S A,..., S D ) The Fault Detection and Identification (FDI) Problem Given random vibration data x1 N, y1 N (x[1],..., x[n]; y[1],..., y[n]) determine whether the structure is healthy or faulty (fault detection). If faulty, determine the fault mode/type (fault identification/localization). Also determine the fault magnitude/size (fault estimation).

7 Introduction FDI process

8 Introduction Classes of time series methods for SHM Methods Pros Cons Non-parametric + simplicity - potentially reduced accuracy + computational efficiency + some user expertise required Parametric + improved parsimony - increased complexity + potentially increased accuracy - computationally intensive - increased user expertise required

9 Non-Parametric methods PSD-based method (response-only case) Main idea FDI is associated with changes in PSD. Q = S YY (ω) = S(ω) 2K Ŝ YY Welch (ω) S YY (ω) χ 2 (2K) ( ) central chi-square distribution with 2K degrees of freedom K : number of segments Fault detection: We set up the statistical hypothesis testing problem H o : S u (ω) = S o (ω) (null hypothesis healthy structure) H 1 : S u (ω) S o (ω) (alternative hypothesis faulty structure) We form the statistic: F = Ŝo(ω)/So(ω) Ŝ u(ω)/s u(ω) F (2K, 2K) (central) F distribution with 2K, 2K degrees of freedom F = X 1/n 1 X 2 /n 2 F (n 1, n 2 ) if : X 1 χ 2 (n 1 ) X 2 χ 2 (n 2 ) and mutually independent

10 Non-Parametric methods Under H o : F = Ŝo(ω) Ŝ u(ω) Hence the proper hypothesis is selected as: F (2K, 2K) at the risk level α (risk level: the probability of accepting H 1 although H o is true false alarm) Remarks 1 A similar procedure may be followed for fault identification. 2 The response signal should be scaled to account for different excitation levels environmental conditions should be constant.

11 Non-Parametric methods FRF-based method (excitation-response case) Main idea FDI is associated with changes in the FRF magnitude (and/or phase). Q = H(jω) (jω) ŜXX Welch (ω) ŜWelch Ĥ(jω) = YX N ( H(jω), σ 2 (ω) ) with σ 2 (ω) 1 γ2 (ω) γ 2 (ω) 2K H(jω) 2 S XX S YX K : PSD function : CSD function : # of non-overlapping segments Fault detection: We set up the statistical hypothesis testing problem H o : H o(jω) H u(jω) = 0 (null hypothesis healthy structure) H 1 : H o(jω) H u(jω) 0 (alternative hypothesis faulty structure) We form the statistic: difference of δ Ĥ(jω) = Ĥo(jω) Ĥu(jω) N(δ H(jω), independent normal δσ2 (ω)) variables

12 Non-Parametric methods FRF-based method (excitation-response case) Under H o : Ĥ(jω) N( 0, 2σ 2 o(ω) ) Hence the proper hypothesis is selected as:

13 Parametric methods Model Parameter based methods (response or excitation-response case) Main idea Fault detection, identification, estimation is associated with changes in the parameter vector θ of a suitable parametric model. Q = f (θ) ˆθ N(θ, Γ) Fault detection: We set up the statistical hypothesis testing problem H o : δθ = θ o θ u = 0 (null hypothesis healthy structure) H 1 : δθ = θ o θ u 0 (alternative hypothesis faulty structure) We form the statistic: δˆθ = ˆθ o ˆθ u N (δθ, δγ) { difference of independent normal variables } Under H o: δˆθ N(0, 2Γ o) and the statistic: Q = δˆθ T δ(2γ o) 1 δˆθ χ 2 (d) χ 2 distribution with d degrees of freedom ( d = dim(θ) )

14 Parametric methods Model Parameter based methods (response or excitation-response case) Hence the proper hypothesis is selected as: Remark Modal models, in which θ consists of modal parameters, may be also used.

15 Parametric methods Residual based methods (response-only or excitation-response case) Main idea Fault detection, identification, estimation is associated with changes in the residual sequences obtained by driving the current signals through predetermined parametric models. Q = f (e[t])

16 Fault Detection and Identification Application Study The Problem Fault detection and identification in structures based on their response vibration measurements. Acceleration (m/s 2 ) Frequency (Hz) Time (s) Specific aims Vibration based fault detection and identification in structures using: a single vibration response measurement, AutoRegressive modelling, statistical decision making.

17 Pick-and-Place Mechanism The Laboratory Setup Exciter Motor A 3 5 Motor B Base 4 Siglab Output Input Siglab Output Conditioner Input DAQs P&P structure: Motor stroke: 19 cm Random excitation via an electromechanical shaker Vertical accelerations measured at 6 locations Base: 110(L) 10(W) 3(H) cm Weight: 14.5 kgr PC

18 The faults Structural State Healthy Fault Type Fault Type Fault Type Fault Type Fault Type Fault Type The total Description A removal of bolt A1 B removal of bolt B1 C removal of bolts C1 and C2 D loosening of motor B slider E loosening of bolt E1 F adding a mass on motor A slider weight of the mechanism is 14.5 kg Added weight (g) Total number of FDI experiments

19 The Nonstationary Experiments Distance (mm) Error (mm) A 2 A 1 B mm 180 mm motor A Distance (mm) B 1 motor B Actual pos. Reference pos. Actual pos. Reference pos. 0 0 A 1 A 2 A 1 B 1 B 2 B Time (s) Error (mm) Time (s) A Single Experiment: In a single experiment the motors move from their rightmost to their leftmost end point and back following a sinus position profile (total time 10 s). The Vibration Signals: Sampling frequency: 512 Hz Bandwidth: Hz N = 5120 samples Analysis based on output 4 A series of 246 experiments are carried out: 40 experiments (+1 baseline) with the structure in its healthy state 40 experiments (+1 baseline) with the structure under faulty state (6 different types)

20 Fault Detection based on nonparametric methods

21 Fault Detection based on nonparametric methods FRF-based method

22 Fault Detection based on nonparametric methods PSD-based method

23 Fault Detection and Identification Framework based on AR models Baseline Phase data acquisition Inspection Phase data acquisition single experiment V: o healthy state A fault type A B fault type B model identification Estimation of (characteristic quantity): : healthy state : fault type A : fault type B current experiment u: designates unknown structural state model identification Estimation of current characteristic quantity statistical decision making NO NO NO??? faulty state YES YES YES healthy state fault type A fault type B

24 Fault Detection and Identification Results Structural State n a Healthy 21 Fault Type A 22 Fault Type B 22 Fault Type C 21 Fault Type D 22 Fault Type E 21 Fault Type F 22 χ 2 statistic Healthy Fault A Fault B Fault C Fault D Fault E Fault F Summary FDI results Test Case Fault Detection False Alarms Missed Faults Healthy Fault A Fault B Fault C Fault D Fault E Fault F 0/40 0/40 0/40 0/40 0/40 0/40 0/40 Fault Identification (misclassifications) Fault A Fault B Fault C Fault D Fault E Fault F 0/240 0/240 0/240 0/240 16/240 18/240 (7.5 %)